Civil Engineering Reference
In-Depth Information
(b)
(a)
Figure 2.1
FE mesh compatibility: (a) incompatible mesh; (b) compatible mesh.
surfaces, and a constrained triangulation can be a mesh of an object with discretised bound-
ary surfaces.
Only the geometrical and topological aspects, and not the functional aspects, of an FE
mesh will be emphasised, and in this sense, there is not much difference between a mesh and
an FE mesh. However, apart from the geometrical aspects, FE meshes are meshes to support
numerical computations, which are stricter in the following aspects:
i. FE meshes have to be compatible, i.e. an edge in 2D and a face in 3D can only be
shared by two elements except for the specially designed elements, which could still
converge without full compatibility, as shown in Figure 2.1.
ii. The shape qualities of the elements in an FE mesh have to be optimised to reduce dis-
cretisation error; in particular, inverted and degenerate elements are not allowed.
iii. The size and shape of the elements have to comply with the specified node spacing
function or metric.
iv. The node numbering and orientation have to be consistent.
v. Some nodal points, edges and faces have to be generated at specified positions.
In the sequel, a mesh simply means an FE mesh, and hence, the above five requirements
will be enforced, and the quality of the elements will be optimised as much as possible.
2.3.5 Structured and unstructured meshes
A mesh is called structured if its connectivity is pre-determined, repeating periodically with
a fixed pattern. In 2D, we have structured rectangular meshes and triangular meshes gen-
erated with the aid of a regular grid, and in 3D, regular hexahedral meshes or tetrahedral
meshes derived from regular hexahedral meshes by a uniform pattern of subdivision of each
hexahedron into five or six tetrahedra are structured meshes. As for unstructured meshes,
the pattern of connectivity is not periodic, and the number of elements connected to a node
varies and is unpredictable; in other words, the node element connection is quite random
and copes with the geometry of the domain and other constraints over the entire mesh, and
the size of the element may also vary quite significantly across the mesh.
2.3.6 Mixed and hybrid meshes
A mixed mesh is a mesh consisting of elements of various types. For instance, a 2D mixed
mesh may contain triangular and quadrilateral elements, and a 3D mixed mesh may have
tetrahedral, hexahedral and pentahedral elements in the mesh. On the other hand, a hybrid
mesh is a mesh consisting of elements of different spatial dimensions; for example, a mesh of
tetrahedral and triangular elements in space is a hybrid mesh (Lie et al. 2001).
